It’s the Law Too — the Laws of Logarithms

Summary:
Do you have trouble remembering the laws of logarithms? Do
you know why you can change log(x)+log(y) to a different form, but not
log(x+y)? This page helps you make sense out of the laws of
logarithms.

See also:
All the laws of logarithms flow
directly out of the laws of exponents. If you feel a bit
unsteady with the laws of exponents, please review
them before going on.

Logarithm? What’s a Logarithm?

A logarithm is just an exponent.

To be specific, the logarithm of a number x to a base
b is just the exponent you put onto b to make the result
equal x. For instance, since 5² = 25, we know that 2 (the
power) is the logarithm of 25 to base 5. Symbolically,
log5(25) = 2.

More generically, if x = by, then we
say that y is “the logarithm of x to the base b”
or “the base-b logarithm of x”. In symbols,
y = logb(x).
Every exponential equation can be rewritten as a logarithmic equation,
and vice versa, just by interchanging the x and y
in this way.

Another way to look at it is that the
logbx
function is defined as the inverse of the
bx function. These two statements express
that inverse relationship, showing how an exponential equation is
equivalent to a logarithmic equation:

x = by
is the same as
y = logbx

Example 1:
1000 = 103 is the same as
3 = log101000.

Example 2:
log381 = ? is the same
as 3? = 81.

It can’t be said too often: a logarithm is nothing more than an
exponent. You can write the above definition
compactly, and show the log as an exponent, by
substituting the second equation into the first to eliminate
y:

Read that as “the logarithm of x in base b is the
exponent you put on b to get x as a result.”

Where Did Logs Come From?

Before
pocket calculators—only
three decades ago, but in
“student years” that’s the age of dinosaurs—the answer
was simple. You needed logs to compute most powers and roots with fair
accuracy; even multiplying and dividing most numbers were
easier with logs. Every decent algebra books had pages and pages of log tables at
the back.

The invention of logs in the early 1600s fueled the scientific
revolution. Back then scientists, astronomers especially, used to spend huge amounts
of time crunching numbers on paper. By cutting the time they spent doing
arithmetic, logarithms effectively gave them a longer productive
life. The slide rule,
once almost a cartoon trademark of a scientist,
was nothing more than a device built for doing various computations
quickly, using logarithms.
See Eli Maor’s e: The Story of a Number for more on
this.

Today, logs are no longer used in routine number crunching.
But there are still good reasons for studying them.

Why Do We Care?

Why do we use logarithms, anyway? I could write a whole article
about them—maybe one day. But for now. ...

To model many natural processes, particularly in living systems.
We perceive loudness of sound as the logarithm of the actual sound
intensity, and dB (decibels) are a logarithmic scale. We also perceive
brightness of light as the logarithm of the actual light energy, and
star magnitudes are measured on a logarithmic scale.

To measure the pH or acidity of a chemical solution.
The pH is the negative logarithm of the concentration of free
hydrogen ions.

To measure earthquake intensity on the Richter scale.

To analyze exponential processes.
Because the log function is the inverse of the exponential
function, we often analyze an exponential curve by means of
logarithms.
Plotting a set of measured points on “log-log” or “semi-log” paper
can reveal such relationships easily.
Applications include cooling of a dead body, growth of bacteria,
and decay of a radioactive isotopes.
The spread of an epidemic in a population often follows a modified
logarithmic curve called a “logistic”.

To solve some forms of area problems in calculus.
(The area under the curve 1/x, between x=1 and x=A, equals
ln A.)

Also in calculus, differentiating a complicated product becomes
much easier if you first take the logarithm.

(Historically, the main reason for teaching
logs in grade school
was to simplify computation, because the log of a multiplication
“downgrades” it to an addition, and the log of a power
expression “downgrades” it to a multiplication. Of course,
with the widespread availability of personal computing devices,
difficulty of computation is no longer a concern, but logs still have
many applications in their own right.)

“Base”ic Facts

From the definition of a log as inverse of an
exponential, you can immediately get some basic facts.
For instance, if you graph y=10x
(or the exponential with any other positive base), you
see that its range is positive reals; therefore the domain of
y=log x (to any base) is the positive reals. In other
words, you can’t take log 0 or log of a negative number.

(Actually, if you’re willing to go outside the reals, you can take
the log of a negative number.
The technique is taught in many trigonometry courses.)

Log of 1, Log Equaling 1

In the same way, you know that the first power of any number is just that number: b1 = b. Again, turn that around to logarithmic form and you have

logbb = 1 for any base b

Example 3:
ln 1 = 0

Example 4:
log55 = 1

Log as Inverse

A log is an exponent because the log function is the
inverse of the exponential function.
The inverse function undoes the effect of the original function. (I’m
not a big fan of most uses the term “cancel” in math, but it does
fit in this situation.)

This means that if you take the log of an exponential (to the same base, of course), you get back to where you started:

logbbx = xfor any base b

This fact lets you evaluate many logarithms without a calculator.

Example 5:
log5125 =
log5(5³) = 3

Example 6:
log10103.16 = 3.16

Example 7:
ln e-kt/2 = -kt/2

What’s “ln”?

Any positive number is suitable as the base of logarithms, but
two bases are used more than any others:

base oflogarithms

symbol

name

10

log(if no base shown)

common logarithm

e

ln

natural logarithm,pronounced “ell-enn” or “lahn”

Natural logs are logs, and follow all the same rules as any other logarithm. Just remember

ln xmeans logex

Why base e? What’s so special about e?
Most of the explanations need some calculus, for instance that
ex is the only function that is both its own integral
and its own derivative or that e has this beautiful definition in
terms of factorials:

e = 1/0! + 1/1! + 1/2! + 1/3! + ...

Numerically, e is about 2.7182818284. It’s irrational (the decimal
expansion never ends and never repeats), and in fact like π it’s
transcendental (no polynomial equation with integer coefficients has
π or e as a root.)

e (like π) crops up in all sorts of
unlikely places, like computations of compound interest.
It would take a book to explain, and
fortunately there is a book, Eli Maor’s e: The Story of a
Number. He also goes into the history of logarithms, and the book is
well worth getting from your library.

Combining Logs with the Same Base

In a minute we’ll look at the various combinations. But first you
might want to know the general principle: logs reduce operations
by one level. Logs turn a multiplication into an addition, a
division into a subtraction, an exponent into a multiplication, and a
radical into a division. Now let’s see why, and look at some examples.

Multiply Numbers, Add Their Logarithms

Multiplying two expressions corresponds to adding their logarithms.
Can we make sense of this?

Example 11:
log5(5x&sup2) is not equal to
2 log5(5x). Be careful with order of operations!
5x² is 5(x²), not (5x)².
log5(5x²) must first be decomposed as the
log
of the product: log55 +
log5(x²). Then the second term can use the
power rule, log5(x²) =
2 log5x. The first term is just 1. Summing up,
log5(5x²) = 1 +
2 log5x.

Divide Numbers, Subtract Their Logarithms

Since division is the opposite of multiplication, and subtraction
is the opposite of addition, it’s not surprising that dividing two
expressions corresponds to subtracting their logs. While we could go
back again to the compact definition, it’s
probably easier to use the two preceding properties.

Changing the Base

To change the log from base b to another base (call it a), you want to find loga(x). Since you already have x on one side of the above equation, it seems like a good start is to take the base-a log of both sides:

loga(blogbx) = logax

But the left-hand side of that equation is just the log of a power. You remember that log(xy) is just log(x) times y. So the equation simplifies to

(logab) (logbx) = logax

Notice that logab is a constant. This
means that the logs of all numbers in a given base a are
proportional to the logs of the same numbers in another base b,
and the proportionality constant logab is
the log of one base in the other base.
If you’re like me, you may have trouble remembering
whether to multiply or divide. If so, just derive the
equation—as you see, it takes only two steps.

Some textbooks present the change-of-base formula as a fraction. To get the fraction from the above equation, simply divide by the proportionality constant logab:

Summary

The laws of logarithms have been scattered through this longish
page, so it might be helpful to collect them in one place. To make
this even more amazingly helpful <grin>, the associated
laws of exponents are shown here too.

For heaven’s sake, don’t try to memorize this table! Just use it to
jog your memory as needed. Better yet, since a log is an exponent, use the
laws of exponents to re-derive any property
of logarithms that you may have forgotten. That way you’ll truly gain
mastery of this material, and you’ll feel confident about the
operations.

exponents

logarithms

(All laws apply for any positive a, b, x, and y.)

x = by
is the same as
y = logbx

b0 = 1

logb1 = 0

b1 = b

logbb = 1

b(logbx) = x

logbbx = x

bxby = bx+y

logb(xy) = logbx + logby

bx÷by = bx−y

logb(x/y) = logbx − logby

(bx)y = bxy

logb(xy) = y logbx

(logab) (logbx) = logax

logbx = (logax) / (logab)

logba = 1 / (logab)

Don’t get creative! Most variations on the above are not valid.

Example 18:
log (5+x) is not the same as
log 5 + log x. As you know,
log 5 + log x = log(5x), not
log(5+x). Look carefully at the above table and you’ll see
that there’s nothing you can do to split up log(x+y) or
log(x−y).

Example 19:
(log x) /
(log y) is not the same as log(x/y). In
fact, when you divide two logs to the same base, you’re
working the change-of-base formula backward. Though it’s not often
useful, (log x) / (log y) =
logyx. Just don’t write log(x/y)!

Example 20:
(log 5)(log x) is not the
same as log(5x). You know that log(5x) is
log 5 + log x. There’s really not much you can
do with the product of two logs when they have the same base.

Conclusion

Well, there you have it: the laws of logarithms
demystified! The general rule is that logs simply drop an operation
down one level: exponents become multipliers, divisions become
subtractions, and so on. If ever you’re unsure of an operation, like
how to change base, work
it out by using the definition of a log and
applying the laws of exponents, and you
won’t go wrong.